432 Class 04

Thomas E. Love, Ph.D.

2024-01-25

Today’s Agenda

  • Fitting two-factor ANOVA/ANCOVA models with lm
    • Incorporating an interaction between factors
    • Incorporating a quantitative covariate
    • Using a quadratic polynomial fit
  • Regression Diagnostics via Residual Plots
  • Validating / evaluating results with yardstick

Appendix

How the c4im data were created from smart_ohio.csv

Today’s R Setup

knitr::opts_chunk$set(comment = NA)

library(janitor)
library(broom)
library(gt)
library(car)
library(mosaic)
library(patchwork)       
library(naniar)
library(simputation)    ## single imputation of missing data
library(rsample)        ## data splitting
library(yardstick)      ## evaluating fits
library(rms)            ## regression tools (Frank Harrell)
library(tidyverse)      

theme_set(theme_bw())

The c4im data

The c4im data

  • 894 subjects in Cleveland-Elyria with bmi and no history of diabetes (missing values singly imputed: assume MAR)
  • All subjects have hx_diabetes (all 0), and are located in the MMSA labeled Cleveland-Elyria.
  • See Course Notes Chapter on BRFSS SMART data for variable details
  • Appendix provides details on data development.

The Five Variables We’ll Use Today

9 variables in the data but we’ll use only these 5 today.

Variable Description
ID subject identifying code
bmi (outcome) Body-Mass index in kg/m2.
exerany any exercise in the past month: 1 = yes, 0 = no
genhealth self-reported overall health (5 levels)
fruit_day average fruit servings consumed per day

Data Load

c4im <- read_rds("c04/data/c4im.Rds")
c4im |> n_miss()
[1] 0
identical(nrow(c4im), n_distinct(c4im$ID))
[1] TRUE

Splitting the Sample

set.seed(432)    ## for future replication
c4im_split <- initial_split(c4im, prop = 3/4)
train_c4im <- training(c4im_split)
test_c4im <- testing(c4im_split)
c(nrow(c4im), nrow(train_c4im), nrow(test_c4im))
[1] 894 670 224

Models We’ll Build Today

  1. Predict bmi using exer_any and genhealth (both categorical)
    • without then with an interaction between the predictors
  2. Add in a quantitative covariate, fruit_day.
  3. Incorporate fruit_day using a quadratic polynomial.

We’ll fit all of these models with lm, and assess them in terms of in-sample (training) fit and out-of-sample (testing) performance.

Consider transforming bmi?

m0 <- lm(bmi ~ exerany + health, data = train_c4im)
boxCox(m0)

Should we transform bmi?

Re-scaling the transformation

bind_rows( favstats(~ 1/bmi, data = train_c4im),
           favstats(~ 1000/bmi, data = train_c4im)) |>
  mutate(outcome = c("1/bmi", "1000/bmi")) |> 
  relocate(outcome) |>
  gt() |> fmt_number(columns = min:sd, decimals = 3) |> 
  tab_options(table.font.size = 20)
outcome min Q1 median Q3 max mean sd n missing
1/bmi 0.016 0.032 0.037 0.042 0.064 0.037 0.008 670 0
1000/bmi 15.873 32.248 36.839 41.806 63.654 37.240 7.606 670 0

Shape doesn’t change

Means by exerany and health

summaries_1 <- train_c4im |>
    group_by(exerany, health) |>
    summarise(n = n(), mean = mean(1000/bmi), stdev = sd(1000/bmi))
summaries_1 
# A tibble: 10 × 5
# Groups:   exerany [2]
   exerany health     n  mean stdev
     <int> <fct>  <int> <dbl> <dbl>
 1       0 E         18  36.9  4.70
 2       0 VG        54  38.6  7.50
 3       0 G         58  34.9  8.51
 4       0 F         31  30.7  8.49
 5       0 P          8  29.7  7.24
 6       1 E         92  39.9  6.50
 7       1 VG       191  38.5  6.87
 8       1 G        152  35.8  7.20
 9       1 F         49  38.2  7.63
10       1 P         17  37.9 10.4

Code for Interaction Plot

ggplot(summaries_1, aes(x = health, y = mean, 
                        col = factor(exerany))) +
  geom_point(size = 2) +
  geom_line(aes(group = factor(exerany))) +
  scale_color_viridis_d(option = "C", end = 0.5) +
  labs(title = "Observed Means of 1000/BMI",
       subtitle = "by Exercise and Overall Health")
  • Note the use of factor here since the exerany variable is in fact numeric, although it only takes the values 1 and 0.
    • Sometimes it’s helpful to treat 1/0 as a factor, and sometimes not.
  • Where is the evidence of serious non-parallelism (if any) in the plot on the next slide that results from this code?

Resulting Interaction Plot

Fitting a Two-Way ANOVA model for 1000/BMI

Model m1 without interaction

m1 <- lm(1000/bmi ~ exerany + health, data = train_c4im)
  • How well does this model fit the training data?
glance(m1) |> 
    select(r.squared, adj.r.squared, sigma, nobs, 
           df, df.residual, AIC, BIC) |> 
  gt() |> fmt_number(columns = r.squared:sigma, decimals = 3) |>
  fmt_number(columns = AIC:BIC, decimals = 1) |>
  tab_options(table.font.size = 20)
r.squared adj.r.squared sigma nobs df df.residual AIC BIC
0.064 0.057 7.385 670 5 664 4,588.7 4,620.2

Tidied ANOVA for m1

tidy(anova(m1)) |> gt() |> 
  fmt_number(columns = sumsq:statistic, decimals = 2) |>
  fmt_number(columns = p.value, decimals = 4) |>
  tab_options(table.font.size = 20)
term df sumsq meansq statistic p.value
exerany 1 859.17 859.17 15.75 0.0001
health 4 1,624.63 406.16 7.45 0.0000
Residuals 664 36,217.09 54.54 NA NA

Tidied summary of m1 coefficients

tidy(m1, conf.int = TRUE, conf.level = 0.90) |> 
  gt() |> fmt_number(columns = estimate:conf.high, decimals = 3) |>
  tab_options(table.font.size = 20)
term estimate std.error statistic p.value conf.low conf.high
(Intercept) 37.593 0.897 41.909 0.000 36.116 39.071
exerany 2.150 0.664 3.237 0.001 1.056 3.245
healthVG −0.725 0.848 −0.855 0.393 −2.123 0.672
healthG −3.588 0.872 −4.112 0.000 −5.025 −2.151
healthF −3.601 1.095 −3.287 0.001 −5.405 −1.796
healthP −3.784 1.640 −2.308 0.021 −6.485 −1.083

Interpreting m1

Name exerany health predicted 1000/bmi
Harry 0 Excellent 37.59
Sally 1 Excellent 37.59 + 2.15 = 39.74
Billy 0 Fair 37.59 - 3.60 = 33.99
Meg 1 Fair 37.59 + 2.15 - 3.60 = 36.14
  • Effect of exerany?
  • Effect of health = Fair instead of Excellent?

m1 Residual Plots (n = 670)

par(mfrow = c(1,2)); plot(m1, which = c(1,2))

m1 Residual Plots (n = 670)

par(mfrow = c(1,2)); plot(m1, which = c(3,5))

Fitting ANOVA model m1int including interaction

Adding the interaction term to m1

m1int <- lm(1000/bmi ~ exerany * health, data = train_c4im)
  • How do our models compare on fit to the training data?
bind_rows(glance(m1), glance(m1int)) |>
  mutate(mod = c("m1", "m1int")) |>
  select(mod, r.sq = r.squared, adj.r.sq = adj.r.squared, 
         sigma, nobs, df, df.res = df.residual, AIC, BIC) |> 
  gt() |> fmt_number(columns = r.sq:sigma, decimals = 3) |>
  fmt_number(columns = AIC:BIC, decimals = 1) |>
  tab_options(table.font.size = 20)
mod r.sq adj.r.sq sigma nobs df df.res AIC BIC
m1 0.064 0.057 7.385 670 5 664 4,588.7 4,620.2
m1int 0.091 0.079 7.301 670 9 660 4,577.2 4,626.8

ANOVA for the m1int model

tidy(anova(m1int)) |> gt() |> 
  fmt_number(columns = sumsq:statistic, decimals = 2) |>
  fmt_number(columns = p.value, decimals = 4) |>
  tab_options(table.font.size = 20)
term df sumsq meansq statistic p.value
exerany 1 859.17 859.17 16.12 0.0001
health 4 1,624.63 406.16 7.62 0.0000
exerany:health 4 1,036.15 259.04 4.86 0.0007
Residuals 660 35,180.94 53.30 NA NA

ANOVA test comparing m1 to m1int

anova(m1, m1int)
Analysis of Variance Table

Model 1: 1000/bmi ~ exerany + health
Model 2: 1000/bmi ~ exerany * health
  Res.Df   RSS Df Sum of Sq      F    Pr(>F)    
1    664 36217                                  
2    660 35181  4    1036.2 4.8596 0.0007223 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

m1int coefficients

tidy(m1int, conf.int = TRUE, conf.level = 0.90) |>
  gt() |> fmt_number(columns = estimate:conf.high, decimals = 3) |>
  tab_options(table.font.size = 20)
term estimate std.error statistic p.value conf.low conf.high
(Intercept) 36.950 1.721 21.472 0.000 34.115 39.784
exerany 2.920 1.882 1.552 0.121 −0.179 6.020
healthVG 1.656 1.987 0.834 0.405 −1.617 4.929
healthG −2.030 1.970 −1.030 0.303 −5.274 1.215
healthF −6.264 2.164 −2.895 0.004 −9.827 −2.700
healthP −7.238 3.102 −2.333 0.020 −12.348 −2.128
exerany:healthVG −2.999 2.192 −1.368 0.172 −6.610 0.612
exerany:healthG −2.033 2.193 −0.927 0.354 −5.646 1.579
exerany:healthF 4.629 2.520 1.837 0.067 0.479 8.779
exerany:healthP 5.256 3.652 1.439 0.151 −0.760 11.272

Interpreting the m1int model

Name exerany health predicted 1000/bmi
Harry 0 Excellent 36.95
Sally 1 Excellent 36.95 + 2.92 = 39.87
Billy 0 Fair 36.95 - 6.26 = 30.69
Meg 1 Fair 36.95 + 2.92 - 6.26 + 4.63 = 38.24
  • How do we interpret effect sizes here? It depends.

Interpreting the m1int model

  • Effect of exerany on predicted 1000/bmi?
    • If health = Excellent, effect is +2.92
    • If health = Fair, effect is (2.92 + 4.63) = +7.55
  • Effect of health = Fair instead of Excellent?
    • If exerany = 0 (no), effect is -6.26
    • If exerany = 1 (yes), effect is (-6.26 + 4.63) = -1.63

Residuals from m1int? (n = 670)

par(mfrow = c(1,2)); plot(m1int, which = c(1,2))

Residuals from m1int? (n = 670)

par(mfrow = c(1,2)); plot(m1int, which = c(3,5))

Incorporating a Covariate into our two-way ANOVA models

Add fruit_day to m1

m2 <- lm(1000/bmi ~ fruit_day + exerany + health, data = train_c4im)
  • How well does this model fit the training data?
bind_rows(glance(m1), glance(m2)) |>
  mutate(mod = c("m1", "m2")) |>
  select(mod, r.sq = r.squared, adj.r.sq = adj.r.squared, 
         sigma, df, df.res = df.residual, AIC, BIC) |> 
  gt() |> fmt_number(columns = r.sq:sigma, decimals = 3) |>
  fmt_number(columns = AIC:BIC, decimals = 1) |>
  tab_options(table.font.size = 20)
mod r.sq adj.r.sq sigma df df.res AIC BIC
m1 0.064 0.057 7.385 5 664 4,588.7 4,620.2
m2 0.075 0.066 7.349 6 663 4,583.1 4,619.2

ANOVA for the m2 model

tidy(anova(m2)) |> gt() |> 
  fmt_number(columns = sumsq:statistic, decimals = 2) |>
  fmt_number(columns = p.value, decimals = 4) |>
  tab_options(table.font.size = 20)
term df sumsq meansq statistic p.value
fruit_day 1 692.07 692.07 12.81 0.0004
exerany 1 697.12 697.12 12.91 0.0004
health 4 1,499.96 374.99 6.94 0.0000
Residuals 663 35,811.73 54.01 NA NA

m2 coefficients

tidy(m2, conf.int = TRUE, conf.level = 0.90) |>
  gt() |> fmt_number(columns = estimate:conf.high, decimals = 3) |>
  tab_options(table.font.size = 20)
term estimate std.error statistic p.value conf.low conf.high
(Intercept) 36.603 0.963 38.007 0.000 35.017 38.189
fruit_day 0.702 0.256 2.739 0.006 0.280 1.124
exerany 1.957 0.665 2.943 0.003 0.862 3.052
healthVG −0.642 0.845 −0.760 0.447 −2.034 0.749
healthG −3.413 0.871 −3.921 0.000 −4.847 −1.979
healthF −3.390 1.093 −3.102 0.002 −5.190 −1.590
healthP −3.803 1.632 −2.331 0.020 −6.491 −1.115

m2 Residuals

par(mfrow = c(1,2)); plot(m2, which = c(1,2))

m2 Residuals

par(mfrow = c(1,2)); plot(m2, which = c(3,5))

Include the interaction term?

m2int <- lm(1000/bmi ~ fruit_day + exerany * health, 
          data = train_c4im)

ANOVA for the m2int model

tidy(anova(m2int)) |> gt() |> 
  fmt_number(columns = sumsq:statistic, decimals = 2) |>
  fmt_number(columns = p.value, decimals = 4) |>
  tab_options(table.font.size = 20)
term df sumsq meansq statistic p.value
fruit_day 1 692.07 692.07 13.14 0.0003
exerany 1 697.12 697.12 13.24 0.0003
health 4 1,499.96 374.99 7.12 0.0000
exerany:health 4 1,110.99 277.75 5.27 0.0003
Residuals 659 34,700.74 52.66 NA NA

m2int coefficients

tidy(m2int, conf.int = TRUE, conf.level = 0.90) |>
  gt() |> fmt_number(columns = estimate:conf.high, decimals = 3) |>
  tab_options(table.font.size = 18)
term estimate std.error statistic p.value conf.low conf.high
(Intercept) 35.962 1.741 20.652 0.000 33.094 38.831
fruit_day 0.766 0.254 3.020 0.003 0.348 1.184
exerany 2.598 1.873 1.387 0.166 −0.488 5.683
healthVG 1.711 1.975 0.866 0.387 −1.543 4.964
healthG −1.920 1.958 −0.981 0.327 −5.146 1.305
healthF −6.205 2.150 −2.885 0.004 −9.747 −2.662
healthP −7.734 3.088 −2.505 0.012 −12.821 −2.648
exerany:healthVG −2.961 2.179 −1.359 0.175 −6.550 0.629
exerany:healthG −1.938 2.180 −0.889 0.374 −5.529 1.653
exerany:healthF 4.867 2.505 1.943 0.052 0.741 8.994
exerany:healthP 5.930 3.637 1.631 0.103 −0.060 11.921

ANOVA: Compare m2 & m2int

anova(m2, m2int)
Analysis of Variance Table

Model 1: 1000/bmi ~ fruit_day + exerany + health
Model 2: 1000/bmi ~ fruit_day + exerany * health
  Res.Df   RSS Df Sum of Sq      F   Pr(>F)    
1    663 35812                                 
2    659 34701  4      1111 5.2747 0.000347 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

m2int Residuals

par(mfrow = c(1,2)); plot(m2int, which = c(1,2))

m2int Residuals

par(mfrow = c(1,2)); plot(m2int, which = c(3,5))

Comparing Our Models

Which of the four models fits best?

In the training sample, we have…

mod r.sq adj.r.sq sigma df df.res AIC BIC
m1 0.064 0.057 7.385 5 664 4,588.7 4,620.2
m2 0.075 0.066 7.349 6 663 4,583.1 4,619.2
m1int 0.091 0.079 7.301 9 660 4,577.2 4,626.8
m2int 0.103 0.090 7.256 10 659 4,570.0 4,624.1
  • Adjusted \(R^2\), \(\sigma\) and AIC all improve as we move down from m1 towards m2_int. BIC likes m2.
  • BUT the training sample cannot judge between models accurately. Our models have already seen that data.

What does augment() give us?

m1_test_aug <- augment(m1, newdata = test_c4im) |> 
  mutate(out = 1000/bmi)
m1_test_aug |> select(ID, bmi, out, .fitted, .resid, health, exerany) |>
  slice(198:202) |> gt() |> 
  fmt_number(columns = bmi:.resid, decimals = 2) |>
  tab_options(table.font.size = 20)
ID bmi out .fitted .resid health exerany
1016 28.44 35.16 33.81 1.35 P 0
1018 26.68 37.48 39.74 −2.26 E 1
1019 25.74 38.85 36.14 2.71 F 1
1020 20.57 48.61 39.02 9.60 VG 1
1024 24.52 40.78 34.01 6.78 G 0

Here, .fitted = predicted out and .resid = out - .fitted.

What to do?

Our models predict 1000/bmi, but we want to assess predictions of bmi. How do we convert predicted 1000/bmi to predicted bmi?

Note that 1000/(1000/bmi) = bmi, so we need

  • 1000/.fitted for our predicted bmi, and
  • observed bmi - predicted bmi for our residuals

Adjusting augment() appropriately

m1_test_aug <- augment(m1, newdata = test_c4im) |> 
  mutate(bmi_fit = 1000/.fitted, bmi_res = bmi - bmi_fit)
m1_test_aug |> 
  select(ID, bmi, bmi_fit, bmi_res, health, exerany, .fitted, .resid) |>
  slice(198:202) |> gt() |> 
  fmt_number(columns = bmi:bmi_res, decimals = 2) |>
  fmt_number(columns = .fitted:.resid, decimals = 2) |>
  tab_options(table.font.size = 20)
ID bmi bmi_fit bmi_res health exerany .fitted .resid
1016 28.44 29.58 −1.14 P 0 33.81 1.35
1018 26.68 25.16 1.52 E 1 39.74 −2.26
1019 25.74 27.67 −1.93 F 1 36.14 2.71
1020 20.57 25.63 −5.06 VG 1 39.02 9.60
1024 24.52 29.41 −4.89 G 0 34.01 6.78

Augment all four models so far…

m1_test_aug <- augment(m1, newdata = test_c4im) |>
  mutate(bmi_fit = 1000/.fitted, bmi_res = bmi - bmi_fit)

m1int_test_aug <- augment(m1int, newdata = test_c4im) |>
  mutate(bmi_fit = 1000/.fitted, bmi_res = bmi - bmi_fit)

m2_test_aug <- augment(m2, newdata = test_c4im) |>
  mutate(bmi_fit = 1000/.fitted, bmi_res = bmi - bmi_fit)

m2int_test_aug <- augment(m2int, newdata = test_c4im) |>
  mutate(bmi_fit = 1000/.fitted, bmi_res = bmi - bmi_fit)

Using the yardstick package

The yardstick package

For each subject in the testing set, we will need:

  • estimate = model’s prediction of that subject’s bmi
  • truth = the bmi value observed for that subject

Calculate a summary of the predictions across the \(n\) test subjects

Summaries from yardstick

  • \(R^2\) = square of the correlation between truth and estimate
  • mae = mean absolute error …

\[ mae = \frac{1}{n} \sum{|truth - estimate|} \]

  • rmse = root mean squared error …

\[ rmse = \sqrt{\frac{1}{n} \sum{(truth - estimate)^2}} \]

Testing Results (Validated \(R^2\))

We can use the yardstick package and its rsq() function.

testing_r2 <- bind_rows(
    rsq(m1_test_aug, truth = bmi, estimate = bmi_fit),
    rsq(m1int_test_aug, truth = bmi, estimate = bmi_fit),
    rsq(m2_test_aug, truth = bmi, estimate = bmi_fit),
    rsq(m2int_test_aug, truth = bmi, estimate = bmi_fit)) |>
    mutate(model = c("m1", "m1int", "m2", "m2int"))
testing_r2 |> 
  gt() |> fmt_number(.estimate, decimals = 3) |>
  tab_options(table.font.size = 20)
.metric .estimator .estimate model
rsq standard 0.078 m1
rsq standard 0.036 m1int
rsq standard 0.069 m2
rsq standard 0.032 m2int

Mean Absolute Error?

Consider the mean absolute prediction error …

testing_mae <- bind_rows(
    mae(m1_test_aug, truth = bmi, estimate = bmi_fit),
    mae(m1int_test_aug, truth = bmi, estimate = bmi_fit),
    mae(m2_test_aug, truth = bmi, estimate = bmi_fit),
    mae(m2int_test_aug, truth = bmi, estimate = bmi_fit)) |>
    mutate(model = c("m1", "m1int", "m2", "m2int"))
testing_mae |> 
  gt() |> fmt_number(.estimate, decimals = 3) |>
  tab_options(table.font.size = 20)
.metric .estimator .estimate model
mae standard 4.296 m1
mae standard 4.463 m1int
mae standard 4.309 m2
mae standard 4.485 m2int

Root Mean Squared Error?

How about the square root of the mean squared prediction error, or RMSE?

testing_rmse <- bind_rows(
   rmse(m1_test_aug, truth = bmi, estimate = bmi_fit),
   rmse(m1int_test_aug, truth = bmi, estimate = bmi_fit),
   rmse(m2_test_aug, truth = bmi, estimate = bmi_fit),
   rmse(m2int_test_aug, truth = bmi, estimate = bmi_fit)) |>
   mutate(model = c("m1", "m1int", "m2", "m2int"))
testing_rmse |> 
  gt() |> fmt_number(.estimate, decimals = 3) |>
  tab_options(table.font.size = 20)
.metric .estimator .estimate model
rmse standard 5.628 m1
rmse standard 5.842 m1int
rmse standard 5.650 m2
rmse standard 5.888 m2int

Other yardstick summaries (1)

  • rsq_trad() = defines \(R^2\) using sums of squares.
    • The rsq() measure we showed a few slides ago is a squared correlation coefficient guaranteed to be in (0, 1).
  • mape() = mean absolute percentage error
  • mpe() = mean percentage error

Other yardstick summaries (2)

  • huber_loss() = Huber loss (often used in robust regression), which is less sensitive to outliers than rmse().
  • ccc() = concordance correlation coefficient, which attempts to measure both consistency/correlation (like rsq()) and accuracy (like rmse()).

See the yardstick home page for more details.

Incorporating Non-Linearity into our models

Polynomial Regression

A polynomial in the variable x of degree D is a linear combination of the powers of x up to D.

For example:

  • Linear: \(y = \beta_0 + \beta_1 x\)
  • Quadratic: \(y = \beta_0 + \beta_1 x + \beta_2 x^2\)
  • Cubic: \(y = \beta_0 + \beta_1 x + \beta_2 x^2 + \beta_3 x^3\)
  • Quartic: \(y = \beta_0 + \beta_1 x + \beta_2 x^2 + \beta_3 x^3 + \beta_4 x^4\)
  • Quintic: \(y = \beta_0 + \beta_1 x + \beta_2 x^2 + \beta_3 x^3 + \beta_4 x^4 + \beta_5 x^5\)

Fitting such a model creates a polynomial regression.

Adding a polynomial in fruit_day

Can we predict 1000/bmi with a polynomial in fruit_day?

lm(1000/bmi ~ fruit_day, data = train_c4im)
lm(1000/bmi ~ poly(fruit_day, 2), data = train_c4im)
lm(1000/bmi ~ poly(fruit_day, 3), data = train_c4im)

Plotting the Polynomials

p1 <- ggplot(train_c4im, aes(x = fruit_day, y = 1000/bmi)) +
    geom_point(alpha = 0.3) + 
    geom_smooth(formula = y ~ x, method = "lm", 
                col = "red", se = FALSE) + 
    labs(title = "Linear Fit")

p2 <- ggplot(train_c4im, aes(x = fruit_day, y = 1000/bmi)) +
    geom_point(alpha = 0.3) + 
    geom_smooth(formula = y ~ poly(x, 2), method = "lm",
                col = "blue", se = FALSE) +
    labs(title = "2nd order Polynomial")

p3 <- ggplot(train_c4im, aes(x = fruit_day, y = 1000/bmi)) +
    geom_point(alpha = 0.3) + 
    geom_smooth(formula = y ~ poly(x, 3), method = "lm",
                col = "purple", se = FALSE) +
    labs(title = "3rd order Polynomial")

p1 + p2 + p3

Plotting the Polynomials

Raw vs. Orthogonal Polynomials

Predict 1000/bmi using fruit_day with a “raw polynomial of degree 2.”

(temp1 <- lm(1000/bmi ~ fruit_day + I(fruit_day^2), 
             data = train_c4im))

Call:
lm(formula = 1000/bmi ~ fruit_day + I(fruit_day^2), data = train_c4im)

Coefficients:
   (Intercept)       fruit_day  I(fruit_day^2)  
       35.4429          1.5095         -0.1201

Predicted 1000/bmi for fruit_day = 2 is

1000/bmi = 35.4429 + 1.5095 (fruit_day) - 0.1201 (fruit_day^2)
         = 35.4429 + 1.5095 (2) - 0.1201 (4)
         = 37.9815

Does the raw polynomial match our expectations?

temp1 <- lm(1000/bmi ~ fruit_day + I(fruit_day^2), 
             data = train_c4im)

augment(temp1, newdata = tibble(fruit_day = 2)) |> 
  gt() |> tab_options(table.font.size = 20)
fruit_day .fitted
2 37.9814

This matches our “by hand” calculation, within rounding error.

  • But it turns out most regression models use orthogonal rather than raw polynomials…

Fitting an Orthogonal Polynomial

Predict 1000/bmi using fruit_day with an orthogonal polynomial of degree 2.

(temp2 <- lm(1000/bmi ~ poly(fruit_day,2), data = train_c4im))

Call:
lm(formula = 1000/bmi ~ poly(fruit_day, 2), data = train_c4im)

Coefficients:
        (Intercept)  poly(fruit_day, 2)1  poly(fruit_day, 2)2  
              37.24                26.31                -9.15

This looks very different from our previous version of the model. What happens when we make a prediction, though?

Prediction in the Orthogonal Polynomial Model

Remember that in our raw polynomial model, our “by hand” and “using R” calculations each predicted 1000/bmi for a subject with fruit_day = 2 to be 37.981.

What happens with the orthogonal polynomial model temp2?

augment(temp2, newdata = data.frame(fruit_day = 2)) |> 
  gt() |> tab_options(table.font.size = 20)
fruit_day .fitted
2 37.9814
  • No change in the prediction.

Fits of raw vs orthogonal polynomials

temp1_aug <- augment(temp1, train_c4im)
temp2_aug <- augment(temp2, train_c4im)

p1 <- ggplot(temp1_aug, aes(x = fruit_day, y = 1000/bmi)) +
    geom_point(alpha = 0.3) +
    geom_line(aes(x = fruit_day, y = .fitted), col = "red", size = 2) +
    labs(title = "temp1: Raw fit, degree 2")

p2 <- ggplot(temp2_aug, aes(x = fruit_day, y = 1000/bmi)) +
    geom_point(alpha = 0.3) +
    geom_line(aes(x = fruit_day, y = .fitted), col = "blue", size = 2) +
    labs(title = "temp2: Orthogonal fit, degree 2")

p1 + p2 + 
    plot_annotation(title = "Comparing Two Methods of Fitting a Quadratic Polynomial")
  • The two models are, in fact, identical.

Fits of raw vs orthogonal polynomials

Why use orthogonal polynomials?

  • The main reason is to avoid having to include powers of our predictor that are highly collinear.
  • Variance Inflation Factor assesses collinearity…
rms::vif(temp1)        ## from rms package
     fruit_day I(fruit_day^2) 
      4.652178       4.652178
  • Orthogonal polynomial terms are uncorrelated…
rms::vif(temp2)      
poly(fruit_day, 2)1 poly(fruit_day, 2)2 
                  1                   1

Why orthogonal polynomials?

The tradeoff is that the raw polynomial is a lot easier to explain in terms of a single equation in the simplest case.

Actually, we’ll usually avoid polynomials in our practical work, and instead use splines, which are more flexible and require less maintenance, but at the cost of pretty much requiring you to focus on visualizing their predictions rather than their equations.

Adding a Second Order Polynomial

m3 <- lm(1000/bmi ~ poly(fruit_day,2) + exerany + health,
          data = train_c4im)
  • Comparison to other models without the interaction…
mod r.squared adj.r.squared sigma df df.residual nobs AIC BIC
m1 0.0642 0.0571 7.385 5 664 670 4,588.7 4,620.2
m2 0.0747 0.0663 7.349 6 663 670 4,583.1 4,619.2
m3 0.0749 0.0651 7.354 7 662 670 4,585.0 4,625.6

m3 coefficients

term estimate std.error statistic p.value conf.low conf.high
(Intercept) 37.655 0.896 42.028 0.000 36.179 39.131
poly(fruit_day, 2)1 20.411 7.439 2.744 0.006 8.158 32.664
poly(fruit_day, 2)2 −2.908 7.529 −0.386 0.699 −15.309 9.493
exerany 1.915 0.674 2.840 0.005 0.804 3.026
healthVG −0.642 0.845 −0.759 0.448 −2.034 0.751
healthG −3.406 0.871 −3.908 0.000 −4.841 −1.970
healthF −3.393 1.093 −3.103 0.002 −5.194 −1.592
healthP −3.723 1.646 −2.263 0.024 −6.434 −1.013

m3 Residuals

par(mfrow = c(1,2)); plot(m3, which = c(1,2))

m3 Residuals

par(mfrow = c(1,2)); plot(m3, which = c(3,5))

Add in the interaction

m3int <- lm(1000/bmi ~ poly(fruit_day,2) + exerany * health,
          data = train_c4im)
  • Comparison to other models with the interaction…
mod r.squared adj.r.squared sigma df df.residual nobs AIC BIC
m1int 0.0910 0.0786 7.301 9 660 670 4,577.2 4,626.8
m2int 0.1034 0.0898 7.256 10 659 670 4,570.0 4,624.1
m3int 0.1034 0.0884 7.262 11 658 670 4,572.0 4,630.6

m3int coefficients

term estimate std.error statistic p.value conf.low conf.high
(Intercept) 37.079 1.713 21.640 0.000 34.257 39.902
poly(fruit_day, 2)1 22.219 7.364 3.017 0.003 10.090 34.348
poly(fruit_day, 2)2 0.216 7.639 0.028 0.977 −12.368 12.799
exerany 2.601 1.878 1.385 0.167 −0.492 5.694
healthVG 1.711 1.977 0.866 0.387 −1.545 4.966
healthG −1.920 1.960 −0.980 0.328 −5.148 1.308
healthF −6.203 2.153 −2.882 0.004 −9.749 −2.658
healthP −7.754 3.168 −2.448 0.015 −12.972 −2.536
exerany:healthVG −2.961 2.181 −1.358 0.175 −6.553 0.631
exerany:healthG −1.940 2.182 −0.889 0.374 −5.534 1.655
exerany:healthF 4.866 2.508 1.940 0.053 0.735 8.997
exerany:healthP 5.951 3.709 1.604 0.109 −0.160 12.061

m3int Residuals

par(mfrow = c(1,2)); plot(m3int, which = c(1,2))

m3int Residuals

par(mfrow = c(1,2)); plot(m3int, which = c(3,5))

Testing Sample for m3 and m3int?

m3_test_aug <- augment(m3, newdata = test_c4im) |>
  mutate(bmi_fit = 1000/.fitted, bmi_res = bmi - bmi_fit)
m3int_test_aug <- augment(m3int, newdata = test_c4im) |>
  mutate(bmi_fit = 1000/.fitted, bmi_res = bmi - bmi_fit)

testing_r2 <- bind_rows(
    rsq(m1_test_aug, truth = bmi, estimate = bmi_fit),
    rsq(m2_test_aug, truth = bmi, estimate = bmi_fit),
    rsq(m3_test_aug, truth = bmi, estimate = bmi_fit),
    rsq(m1int_test_aug, truth = bmi, estimate = bmi_fit),
    rsq(m2int_test_aug, truth = bmi, estimate = bmi_fit),
    rsq(m3int_test_aug, truth = bmi, estimate = bmi_fit)) |>
    mutate(mod = c("m1", "m2", "m3", "m1int", "m2int", "m3int"))
  • I’ve hidden my calculations for RMSE and MAE here.

Test Results for all six models

bind_cols(testing_r2 |> select(mod, rsquare = .estimate), 
          testing_rmse |> select(rmse = .estimate),
          testing_mae |> select(mae = .estimate)) |> 
  mutate(elements = c("exerany + health", "add fruit", "add polynomial", "m1 + interaction", "m2 + interaction", "m3 + interaction")) |>
  gt() |> fmt_number(columns = rsquare:mae, decimals = 4) |>
  tab_options(table.font.size = 20)
mod rsquare rmse mae elements
m1 0.0779 5.6277 4.2962 exerany + health
m2 0.0692 5.6497 4.3087 add fruit
m3 0.0698 5.6477 4.3052 add polynomial
m1int 0.0358 5.8417 4.4628 m1 + interaction
m2int 0.0318 5.8878 4.4850 m2 + interaction
m3int 0.0317 5.8887 4.4855 m3 + interaction
  • Did the polynomial in m3 and m3int improve predictions?

Next Week

  • Fitting splines, as well as polynomial terms.
  • Using the ols function from the rms package to fit linear regression models with non-linear terms.
  • Be sure to submit Lab 2 to Canvas by Tuesday 2024-01-30 at Noon.

Appendix

Creating Today’s Data Set

url1 <- "https://raw.githubusercontent.com/THOMASELOVE/432-data/master/data/smart_ohio.csv"

smart_ohio <- read_csv(url1)

c4 <- smart_ohio |>
    filter(hx_diabetes == 0, 
           mmsa == "Cleveland-Elyria",
           complete.cases(bmi)) |>
    select(bmi, inc_imp, fruit_day, drinks_wk, 
           female, exerany, genhealth, race_eth, 
           hx_diabetes, mmsa, SEQNO) |>            
    type.convert(as.is = FALSE) |>                       
    mutate(ID = as.character(SEQNO - 2017000000)) |>
    relocate(ID)

Codebook for useful c4 variables (1)

  • 894 subjects in Cleveland-Elyria with bmi and no history of diabetes
Variable Description
bmi (outcome) Body-Mass index in kg/m2.
inc_imp income (imputed from grouped values) in $
fruit_day average fruit servings consumed per day
drinks_wk average alcoholic drinks consumed per week
female sex: 1 = female, 0 = male

Codebook for useful c4 variables (2)

  • 894 subjects in Cleveland-Elyria with bmi and no history of diabetes
Variable Description
exerany any exercise in the past month: 1 = yes, 0 = no
genhealth self-reported overall health (5 levels)
race_eth race and Hispanic/Latinx ethnicity (5 levels)
  • plus ID, SEQNO, hx_diabetes (all 0), MMSA
  • See Course Notes Chapter on BRFSS SMART data

Basic Data Summaries

Available approaches include:

  • summary
  • mosaic package’s inspect()
  • Hmisc package’s describe

all of which can work nicely in an HTML presentation, but none of them fit well on one of these slides.

Quick Histogram of each quantitative variable

Code for previous slide

p1 <- ggplot(c4, aes(x = bmi)) + 
    geom_histogram(fill = "navy", col = "white", bins = 20)
p2 <- ggplot(c4, aes(x = inc_imp)) + 
    geom_histogram(fill = "forestgreen", col = "white", 
                   bins = 20)
p3 <- ggplot(c4, aes(x = fruit_day)) + 
    geom_histogram(fill = "tomato", col = "white", bins = 20)
p4 <- ggplot(c4, aes(x = drinks_wk)) + 
    geom_histogram(fill = "royalblue", col = "white", 
                   bins = 20)
(p1 + p2) / (p3 + p4)

I also used #| warning: false in the plot’s code chunk label to avoid warnings about missing values, like this one for inc_imp:

Warning: Removed 120 rows containing non-finite values

Binary variables in raw c4

c4 |> tabyl(female, exerany) |> adorn_title()
        exerany        
 female       0   1 NA_
      0      95 268  20
      1     128 361  22
  • female is based on biological sex (1 = female, 0 = male)
  • exerany comes from a response to “During the past month, other than your regular job, did you participate in any physical activities or exercises such as running, calisthenics, golf, gardening, or walking for exercise?” (1 = yes, 0 = no, don’t know and refused = missing)
  • Any signs of trouble here?

Multicategorical genhealth in raw c4

c4 |> tabyl(genhealth)
   genhealth   n     percent valid_percent
 1_Excellent 148 0.165548098    0.16573348
  2_VeryGood 324 0.362416107    0.36282195
      3_Good 274 0.306487696    0.30683091
      4_Fair 112 0.125279642    0.12541993
      5_Poor  35 0.039149888    0.03919373
        <NA>   1 0.001118568            NA
  • The variable is based on “Would you say that in general your health is …” using the five specified categories (Excellent -> Poor), numbered for convenience after data collection.
  • Don’t know / not sure / refused treated as missing.
  • How might we manage this variable?

Changing the levels for genhealth

c4 <- c4 |>
    mutate(health = 
               fct_recode(genhealth,
                          E = "1_Excellent",
                          VG = "2_VeryGood",
                          G = "3_Good",
                          F = "4_Fair",
                          P = "5_Poor"))

Might want to run a sanity check here, just to be sure…

Checking health vs. genhealth in c4

c4 |> tabyl(genhealth, health) |> adorn_title()
             health                   
   genhealth      E  VG   G   F  P NA_
 1_Excellent    148   0   0   0  0   0
  2_VeryGood      0 324   0   0  0   0
      3_Good      0   0 274   0  0   0
      4_Fair      0   0   0 112  0   0
      5_Poor      0   0   0   0 35   0
        <NA>      0   0   0   0  0   1
  • OK. We’ve preserved the order and we have much shorter labels. Sometimes, that’s helpful.

Multicategorical race_eth in raw c4

c4 |> count(race_eth)
# A tibble: 6 × 2
  race_eth                     n
  <fct>                    <int>
1 Black non-Hispanic         167
2 Hispanic                    27
3 Multiracial non-Hispanic    19
4 Other race non-Hispanic     22
5 White non-Hispanic         646
6 <NA>                        13

“Don’t know”, “Not sure”, and “Refused” were treated as missing.

  • What is this variable actually about?
  • What is the most common thing people do here?

What is the question you are asking?

Collapsing race_eth levels might be rational for some questions.

  • We have lots of data from two categories, but only two.
  • Systemic racism affects people of color in different ways across these categories, but also within them.

Is combining race and Hispanic/Latinx ethnicity helpful?

It’s hard to see the justice in collecting this information and not using it in as granular a form as possible, though this leaves some small sample sizes. There is no magic number for “too small a sample size.”

  • Most people identified themselves in one of the categories.
  • These data are not ordered, and (I’d argue) ordering them isn’t helpful.
  • Regression models are easier to interpret, though, if the “baseline” category is a common one.

Resorting the factor for race_eth

Let’s sort all five levels, from most observations to least…

c4 <- c4 |>
    mutate(race_eth = fct_infreq(race_eth))

c4 |> tabyl(race_eth)
                 race_eth   n    percent valid_percent
       White non-Hispanic 646 0.72259508    0.73325766
       Black non-Hispanic 167 0.18680089    0.18955732
                 Hispanic  27 0.03020134    0.03064699
  Other race non-Hispanic  22 0.02460850    0.02497162
 Multiracial non-Hispanic  19 0.02125280    0.02156640
                     <NA>  13 0.01454139            NA
  • Not a perfect solution, certainly, but we’ll try it out.

“Cleaned” Data and Missing Values

c4 <- c4 |>
    select(ID, bmi, inc_imp, fruit_day, drinks_wk, 
           female, exerany, health, race_eth, everything())

miss_var_summary(c4)
# A tibble: 13 × 3
   variable    n_miss pct_miss
   <chr>        <int>    <dbl>
 1 inc_imp        120   13.4  
 2 exerany         42    4.70 
 3 fruit_day       41    4.59 
 4 drinks_wk       39    4.36 
 5 race_eth        13    1.45 
 6 health           1    0.112
 7 genhealth        1    0.112
 8 ID               0    0    
 9 bmi              0    0    
10 female           0    0    
11 hx_diabetes      0    0    
12 mmsa             0    0    
13 SEQNO            0    0

Single Imputation Approach?

set.seed(43203)
c4im <- c4 |>
    select(ID, bmi, inc_imp, fruit_day, drinks_wk, 
           female, exerany, health, race_eth) |>
    data.frame() |>
    impute_cart(health ~ bmi + female) |>
    impute_pmm(exerany ~ female + health + bmi) |>
    impute_rlm(inc_imp + drinks_wk + fruit_day ~ 
                   bmi + female + health + exerany) |>
    impute_cart(race_eth ~ health + inc_imp + bmi) |>
    tibble()

prop_miss_case(c4im)
[1] 0

Saving the tidied data

Let’s save both the unimputed and the imputed tidy data as R data sets.

write_rds(c4, "c04/data/c4.Rds")

write_rds(c4im, "c04/data/c4im.Rds")

To reload these files, we’ll use read_rds().

  • The main advantage here is that we’ve saved the whole R object, including all characteristics that we’ve added since the original download.

432 Class 04 | 2024-01-25 | https://thomaselove.github.io/432-2024/